 I'm going to check whether I can share it. Yes, can you see my screen now? Yes, everything seems to be okay. Okay, good. First, I would like to thanks to all of you, especially the organizer, to having me here. I will speak about the thermodynamic in non-equilibrium jump process. Okay, I'm Faize, and I'm working with Professor Christian Maus, my supervisor, and this is joint work with Christian Maus and Carol Nitoch-Ney. The contents that I will speak about that first, I will give you an introduction that what is the colorimetry in equilibrium, what is the third law in equilibrium, and then we will go to non-equilibrium colorimetry. I will give you some main ingredients that we need. And then as an example, we will see what the contents are. And finally, the third law of thermodynamic in non-equilibrium. This slide, I will not use the notation, but it's just colorimetry in equilibrium. Just remember that, okay, we have first law of thermodynamic, we have entropy. There is a definition for heat capacity. Of course, it's not always for volume fix, but okay, it's dependent of temperature and heat. Just some reminder, but we will not use it. Third law of thermodynamic is actually Nernst heat theorem. And it has different versions, but the version and the concept, the result of that, that is really important. It's telling that when a temperature is going to zero, the heat capacity also is going to zero. But the Nernst heat theorem at the first, it wasn't as a law, I mean, because it's not universal. And people could not accept it as a third law of thermodynamic, but of course nowadays it's accepted. And it has a condition and it's not a universal, same as the first law of thermodynamic and what is the condition. In some cases, it's satisfying. And when the ground state is non-degenerate, it's satisfying. And if they generate, it's not satisfying. And it has a real example in life. And spin ice is one of the material that it's not satisfying the third law of thermodynamic in equilibrium. Okay, but it's non-equilibrium colorimetry. I will give you two important ingredients. First one is quasi-potential. Just consider, this is a cartoon, it's not simulation. Just consider a heat pass, just consider water, a big container of water. And you are heating that with the joule heating. And first of all, I mean, it's, I'm telling that it's a big continuum because we can fix the temperature on that. And first of all, we have in, this is open system, it's non-equilibrium. And we are in a steady state, non-equilibrium steady state. And then we perturb it in the temperature. Okay, we can perturb in another parameter, but we consider the temperature. And then we again are in a new, after a lot of fluctuation, we are in a new steady non-equilibrium. And we measure in each state, we measure the mean dissipated power. The time integral of these differences of these two is quasi-potential. That you can see that we define it like that. And we are hoping, and we put it like that, that it's going to, I mean, this is the integral is well defined. It means that in the time going to infinity, this one going to, I mean, the limit of this one. And if you want to see that, what does it mean exactly in equilibrium, this quasi-potential is in terms of the energy of each state minus of the average. Okay, the quasi-potential was their main ingredient. And the other one is the definition of the heat capacity in terms of quasi-potential. A discretized inverse of temperature. The derivative of temperature for, I mean, the quasi-potential, we give you the heat capacity. And because the mean value of the V, quasi-potential is zero, you can write it like that. I will not describe everything. You can see the details in non-equilibrium color in the paper with Kirstian Mauss and Karel. And now, really fast, we cover an example. Just consider a quantum switch that with the rate alpha, it's switching, I mean, it's two level system with the rate alpha, it's just changing. And whenever it's changing the state from ground state to excited state, it's giving, I mean, it's releasing heat and exchanging the heat to the environment. And we define it as a Markov jump process. You can model it in a four state graph. And, okay, the main idea is that we can measure the heat capacity as we define it in previous slide. When it's in equilibrium, the heat capacity for different alpha is the plot is like that, this is a standard heat, I mean, plot of the heat capacity, you can just search it and see that in equilibrium, heat capacity is like that. It has a very well-known peak. And then it's going to zero in zero temperature as a third law. And if it's not equilibrium and you are breaking data balance, then you are going to non-equilibrium and the heat capacity, it can be negative as well. I'm not just speaking why and when it's negative. It's another purpose, but it's at the point of this example is that we can measure, we can calculate the heat capacity in a non-equilibrium. Okay, and what is the main theorem? The second law in non-equilibrium is telling that heat capacity is going to zero as well, the heat capacity that we define. And I just opened this to mean it's a nothing special and it's just that they are equal to each other. We want to tell that in two condition, this heat capacity when temperature is going to zero, it's going to zero. Just remember that in equilibrium, we have one condition and now in non-equilibrium we have two conditions. And what are our two conditions? Okay, we have two conditions. The first one is same as equilibrium. It's telling that the ground estate must be the unique, the dominant estate must be in the low temperature, must be the unique dominant estate. And the other one is telling that all of the estate must be accessible to each other, I mean, enough accessible, but what does it mean enough? I mean, we have a mathematical expression for that and we exactly know when it's satisfying and when not. But the intuition is that, for example, here, okay, I will describe it exactly in the next slide, but just remember we need accessibility and we need that one unique dominant estate. And what is the idea? The idea is that if you need one dominant estate, then this row as a stationary distribution can be zero, zero, because everything is focused in one place. And if V is bounded as a quasi-potential, then mathematically it's going to zero as well. This accessibility is speaking about the boundedness of quasi-potential. I think that's the meaning of that. Yes. Imagine the dominant estate in this graph is here and then every estate wants to go here and if there is just one pass from here to here and it's difficult to pass here, it's not enough accessible. And if you add one age, then you give the chance to be more accessible. Okay, in the graph, if you have more ages, it means that they are more accessible together, but we will see the meaning of that. To know what is the, why the row, when it's a unique dominant estate is going to zero and what's in low temperature, how it's looked like, you can see these papers for the low temperature. And now look at this example. The energy for these states, energy 423, and if the energy of U is 6, it's different from the time that it's 10, for example. We can easily calculate it via matrix first theorem as well and we are every approach that the dominant estate is this one. And then for this estate particle want to go from here, I mean from here to there. And if energy is 10, it needs more work to do, to pass from this estate to go here. And it's the only way that must pass is this way. And quasi-potential is not bonded here. But if U is 6, it's compatible with the dominant estate, then it's reasonable and it's bonded. Another way that if we add one edge as a tunerling, okay, it can possible we have tunerling, then we are adding to this, I mean we are giving chance for this accessibility. So the third law of the thermodynamic and non-equilibrium has two conditions. Under these two conditions, the heat capacity is going to zero. The first condition is that in the law of temperature, you need only one dominant estate. And the second thing is to speak in about the accessibility. If all of the assets are more enough accessible to each other, then you will be sure that heat capacity is going to zero. There are the papers that, okay, the concepts and the physical meaning and more, I mean, mathematical, a lot of mathematical relation, you can see in this paper. And all of the mathematical proof that it's under, I mean, it's via graphical representation. It's in this paper. But what does it mean of graphical representation? I gave you a mark of jump process in this slide. Yes, we are defining a mark of jump process. And then whatever we have, it depends on the graph. And the main idea for quasi-potential, just for those who are interested for a section in the distribution is via matrix theory theorem. And for quasi-potential is via matrix forest theorem. And then, yes, I've done. Yes, thanks. Thank you. Yes, I'm ready for the question. Yeah, we have time for a question or two. I don't see you and it's really strange.